Planar massless fermions in Coulomb and Aharonov-Bohm potentials

TL;DR

This paper constructs solutions to the Dirac equation for massless fermions in 2+1 dimensions under Coulomb and Aharonov-Bohm potentials, analyzing virtual bound states and their implications for quantum electrodynamics vacuum restructuring.

Contribution

It provides a detailed analysis of self-adjoint extensions of the Dirac Hamiltonian and the emergence of virtual bound states in this context, which is a novel exploration.

Findings

Virtual bound states appear in overcritical Coulomb regimes.

Derived equations for energies and lifetimes of virtual states.

Analyzed local density of states under combined potentials.

Abstract

Solutions to the Dirac equation are constructed for a massless charged fermion in Coulomb and Aharonov--Bohm potentials in 2+1 dimensions. The Dirac Hamiltonian on this background is singular and needs a one-parameter self-adjoint extension, which can be given in terms of self-adjoint boundary conditions. We show that the virtual (quasistationary) bound states emerge in the presence of an attractive Coulomb potential when the so-called effective charges become overcritical and discuss a restructuring of the vacuum of the quantum electrodynamics when the virtual bound states emerge. We derive equations, which determine the energies and lifetimes of virtual bound states, find solutions of obtained equations for some values of parameters as well as analyze the local density of states as a function of energy in the presence of Coulomb and Aharonov--Bohm potentials.